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First, it is not clear from the physiology or biochemical information alone to determine what evolved from what. The intermediate pyruvate is always there, so lactose dehydrogenase (LDH) and pyruvate decarboxylase (PDC) could have each evolved anytime, and there are good reasons for both:

Add to it that there are organisms like Sch. pombe that have both enzymes, so can do both reactions, your questions should be rather:

How has the enzyme LDH evolved?

The enzyme is similar to malate dehydrogenase, both form a family. They belong to a group of enzymes that all have a NAD(P) binding domain, so it's not too far-fetched to state that LDH and MDH evolved from another enzyme with NAD(P) binding domain.

Protein sequences of pyruvate decarboxylase (PDC) derived from cloned
yeast (Saccharomyces cerevisiae) and bacterial (Zymomonas mobilis)
genes were compared with each other and with sequence databases.
Extensive sequence similarities were found between them and with two
others: cytochrome-linked pyruvate oxidase from Escherichia coli and
acetolactate synthase (ilvI in E. coli; ILV2 gene in S. cerevisiae).
All catalyse decarboxylation of pyruvate using thiamine pyrophosphate
(TPP) as cofactor. General overall similarity suggests common ancestry
for these enzymes.